Data indicate that certain important types of marine organisms behave acoustically like weakly scattering fluid bodies (i.e., their material properties appear fluidlike and similar to those of the surrounding fluid medium). Use of this boundary condition, along with certain assumptions, allows reduction of what is a very complex scattering problem to a relatively simple, approximate ray-based solution. Because of the diversity of this problem, the formulation is presented in two articles: this first one in which the basic physics of the scattering process is described where the incident sound wave is nearly normally incident upon a single target (i.e., the region in which the scattering amplitude is typically at or near a maximum value for the individual) and the second one [Stanton et al., J. Acoust. Soc. Am. 94, 3463–3472 (1993)] where the formulation is heuristically extended to all angles of incidence and then statistically averaged over a range of angles and target sizes to produce a collective echo involving an aggregation of randomly oriented different sized scatterers. In this article, a simple ray model is employed in the deformed cylinder formulation [Stanton, J. Acoust. Soc. Am. 86, 691–705 (1989)] to describe the scattering by finite length deformed fluid bodies in the general shape of elongated organisms. The work involves single realizations of the length and angle of orientation. Straight and bent finite cylinders and prolate spheroids are treated in separate examples. There is reasonable qualitative comparison between the structure of the data collected by Chu et al. [ICES J. Mar. Sci. 49, 97–106 (1992)] involving two decapod shrimp and this single-target normal-incidence theory. This analysis forms the basis for successful comparison (presented in the companion article) between the extended formulation that is averaged over an ensemble of realizations of length and angle of orientation and scattering data involving aggregations of up to 100's of animals.